Lensmaker's Equation Calculator

What is the Lensmaker's Equation?

The Lensmaker's Equation is a formula used in optics to calculate the focal length of a thin lens based on its geometry and the refractive index of the lens material. It relates the focal length of a lens to its radii of curvature and the refractive index of the material from which it is made. This equation is fundamental in lens design and optical system analysis.

Formula

The Lensmaker's Equation is given by:

\[ \frac{1}{f} = (n - 1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right) \]

Where:

• \( f \) is the focal length of the lens (in meters, m)
• \( n \) is the refractive index of the lens material (dimensionless)
• \( R_1 \) is the radius of curvature of the first lens surface (in meters, m)
• \( R_2 \) is the radius of curvature of the second lens surface (in meters, m)

Calculation Steps

Let's calculate the focal length of a lens using the Lensmaker's Equation:

1. Given:
   • Refractive index (\( n \)) = 1.5
   • Radius of curvature 1 (\( R_1 \)) = 0.2 m
   • Radius of curvature 2 (\( R_2 \)) = -0.3 m (negative sign indicates concave surface)
2. Apply the Lensmaker's Equation: \[ \frac{1}{f} = (n - 1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right) \]
3. Substitute the known values: \[ \frac{1}{f} = (1.5 - 1)\left(\frac{1}{0.2} - \frac{1}{-0.3}\right) \]
4. Simplify: \[ \frac{1}{f} = 0.5\left(5 + 3.33\right) = 0.5(8.33) = 4.165 \]
5. Calculate the focal length: \[ f = \frac{1}{4.165} \approx 0.24 \text{ m} \]

Example and Visual Representation

Let's visualize a convex-concave lens and its focal point:

This diagram illustrates:

• A convex-concave lens (blue curves)
• The optical axis (green dashed line)
• The focal point F (red dot)
• The radii of curvature R₁ and R₂